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XI. On the Form of the Cells of Bees. By J. W. L. Glaisher, 



B.A., Fellow of Trinity College, Cambridge*. 



THE following account of the history of the problem of the 

 bee-cell is extracted from pp. 428-430 of l Homes with- 

 out Hands/ by the Rev. J. G. Wood (London, 1869) :— 



" If a single cell be isolated, it will be seen that the sides rise 

 from the outer edges of the three lozenges above mentioned ; so 

 that there are, of course, six sides, the transverse section of which 

 gives a perfect hexagon. Many years ago Maraldi, being struck 

 with the fact that the lozenge-shaped plates always had the same 

 angles, took the trouble to measure them, and found that in 

 each lozenge the large angles measured 109° 28', and the smaller 

 70° 32', the two together making 180°, the equivalent of two 

 right angles. He also noted the fact that the apex of the three- 

 sided cup was formed by the union of three of the greater angles. 

 " Some time afterwards, Reaumur, thinking that this remark- 

 able uniformity of angle might have some connexion with the 

 wonderful economy of space which is observable in the bee-comb, 

 hit upon a very ingenious plan. Without mentioning his rea- 

 sons for the question, he asked Koenig, the mathematician, to 

 make the following calculation : — Given a hexagonal vessel 

 terminated by three lozenge-shaped plates, what are the angles 

 which would give the greatest amount of space with the least 

 amount of material ? Kcenig made his calculations, and found 

 that the angles were 109° 26' and 70° 34', almost precisely 

 agreeing with the measurements of Maraldi. The reader is re- 

 quested to remember these angles. Reaumur, on receiving the 

 answer, concluded that the bee had very nearly solved the diffi- 

 cult mathematical problem, the difference between the measure- 

 ment and the calculation being so small as to be practically 

 negatived in the actual construction of so small an object as the 

 bee-cell. 



" Mathematicians were naturally delighted with the result of 

 the investigation ; for it showed how beautifully practical science 

 could be aided by theoretical knowledge ; and the construction 

 of the bee- cell became a famous problem in the economy of 

 nature. In comparison with the honey which the cell is in- 

 tended to contain, the wax is a rare and costly substance, secreted 

 in very small quantities, and requiring much time for its pro- 

 duction ; it is therefore essential that the quantity of wax em- 

 ployed in making the comb should be as little, and that of the 

 honey contained in it as great as possible. 



" For a long time these statements remained uncontro verted. 



* Communicated bv the Author. 



